Let $\Omega\subset \mathbb{R}^n$ an open contractible set (we can assume $n=2$ for a start) and $\omega$ be a 1-form on $\Omega$ which is nowhere zero. Then $\omega=df$ for a function $f$ if and only if $d\omega=0$. If $\omega$ is not closed it might still be possible that it works up to a positive multiple. In other words, do there exist functions $f,g$ on $\Omega$, $g>0$ such that $df=g\omega$?

For a contractible $\Omega$ this is equivalent to $dg\wedge \omega+gd\omega=0$, and if $h$ denotes the logarithm of $g$, the question is whether there exists $h$ such that $dh\wedge \omega+d\omega=0$. However, this equation does not tell me much.

If the machinery of differential topology has an easy answer to this question I'm also interested what can be said in more general cases (first of all a ring-shaped $\Omega$, or more than 2 dimensions).

This requires a local analysis before you can bring in differential topology. Also, note that dimension 2 is different from higher dimensions, because in dimension 2 there is only 1 equation and the differential of that equation is identically zero, whereas in higher dimensions, there are $n(n-1)/2$ equations and the differential of the equation, $dh\wedge d\omega = 0$ is another nontrivial equation that must also be satisfied. In other words, in higher dimensions the PDE is overdetermined.
–
Deane YangDec 5 '11 at 14:55

1 Answer
1

Well, the local condition that $\omega\not=0$ be a nonzero multiple of an closed $1$-form is that $\omega\wedge d\omega = 0$. This is necessary and sufficient for the local existence of functions $f$ and $g\not=0$ such that $\omega = g\ df$. (This claim is just a special case of the Frobenius Theorem.)

For the global question, you are really asking whether a codimension $1$ foliation of a contractible open set is always the level sets of a function without critical points. This is certainly false in dimensions $3$ and higher, and I sort of remember that it's false in dimension $2$ as well, but I can't remember the example. (It is already false in dimension $2$. I looked it up later; see the added remark below.)

Added remark: Your question is addressed by Exercises 5 and 6 of Section 16 of Chapter XVIII of Volume IV of Dieudonné's Treatise on Analysis. He gives the above local criterion and a counterexample to its global analog with an example of a $1$-form $\omega$ on $\mathbb{R}^2$ that is nonvanishing and yet cannot be written globally in the form $g\ df$ for two smooth functions on $\mathbb{R}^2$. Just to save you the trouble of looking it up, here is his example
$$
\omega = y^3(1{-}y)^2\ dx + \big(y^3-2(1{-}y)^2\bigr)\ dy.
$$
The point is that , if you could write $\omega = g\ df$, then, away from the lines $y=0$ and $y=1$, the function $f$ would have to be a function of
$$
F = x + \frac{1}{y^2} + \frac{1}{1-y}
$$
and $f$ would have to be constant on the lines $y=0$ and $y=1$. Now, you need to check that you can't rig an $f$ with these properties that is smooth and without critical points on the whole plane.